Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [8325,2,Mod(1,8325)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("8325.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(8325, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 8325 = 3^{2} \cdot 5^{2} \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8325.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [6,-2,0,8,0,0,4,-12,0,0,4,0,6] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(66.4754596827\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.6.95034688.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 2x^{5} - 8x^{4} + 12x^{3} + 16x^{2} - 12x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 1665)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(2.28339\) of defining polynomial
Character \(\chi\) \(=\) 8325.1

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.28339 q^{2} +3.21388 q^{4} +3.72551 q^{7} -2.77177 q^{8} +4.52232 q^{11} -4.87523 q^{13} -8.50679 q^{14} -0.0987285 q^{16} +0.0939399 q^{17} +7.35707 q^{19} -10.3262 q^{22} -1.41768 q^{23} +11.1321 q^{26} +11.9733 q^{28} -6.50679 q^{29} +5.98447 q^{31} +5.76897 q^{32} -0.214502 q^{34} +1.00000 q^{37} -16.7991 q^{38} +0.274494 q^{41} +7.48354 q^{43} +14.5342 q^{44} +3.23713 q^{46} -6.61626 q^{47} +6.87940 q^{49} -15.6684 q^{52} -10.4743 q^{53} -10.3262 q^{56} +14.8576 q^{58} +0.507456 q^{59} -2.96122 q^{61} -13.6649 q^{62} -12.9754 q^{64} +7.08320 q^{67} +0.301912 q^{68} -7.51096 q^{71} +1.90544 q^{73} -2.28339 q^{74} +23.6448 q^{76} +16.8479 q^{77} +0.537233 q^{79} -0.626777 q^{82} +7.52649 q^{83} -17.0879 q^{86} -12.5348 q^{88} +0.259584 q^{89} -18.1627 q^{91} -4.55627 q^{92} +15.1075 q^{94} +16.8442 q^{97} -15.7084 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 2 q^{2} + 8 q^{4} + 4 q^{7} - 12 q^{8} + 4 q^{11} + 6 q^{13} - 6 q^{14} + 16 q^{16} - 8 q^{17} + 16 q^{19} + 10 q^{22} - 10 q^{23} + 12 q^{26} + 12 q^{28} + 6 q^{29} + 14 q^{31} - 36 q^{32} + 2 q^{34}+ \cdots + 6 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\). You can download additional coefficients here.



Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.28339 −1.61460 −0.807301 − 0.590140i \(-0.799073\pi\)
−0.807301 + 0.590140i \(0.799073\pi\)
\(3\) 0 0
\(4\) 3.21388 1.60694
\(5\) 0 0
\(6\) 0 0
\(7\) 3.72551 1.40811 0.704054 − 0.710146i \(-0.251371\pi\)
0.704054 + 0.710146i \(0.251371\pi\)
\(8\) −2.77177 −0.979968
\(9\) 0 0
\(10\) 0 0
\(11\) 4.52232 1.36353 0.681766 − 0.731570i \(-0.261212\pi\)
0.681766 + 0.731570i \(0.261212\pi\)
\(12\) 0 0
\(13\) −4.87523 −1.35214 −0.676072 − 0.736835i \(-0.736319\pi\)
−0.676072 + 0.736835i \(0.736319\pi\)
\(14\) −8.50679 −2.27354
\(15\) 0 0
\(16\) −0.0987285 −0.0246821
\(17\) 0.0939399 0.0227838 0.0113919 − 0.999935i \(-0.496374\pi\)
0.0113919 + 0.999935i \(0.496374\pi\)
\(18\) 0 0
\(19\) 7.35707 1.68783 0.843914 − 0.536478i \(-0.180246\pi\)
0.843914 + 0.536478i \(0.180246\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) −10.3262 −2.20156
\(23\) −1.41768 −0.295608 −0.147804 − 0.989017i \(-0.547220\pi\)
−0.147804 + 0.989017i \(0.547220\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 11.1321 2.18318
\(27\) 0 0
\(28\) 11.9733 2.26275
\(29\) −6.50679 −1.20828 −0.604141 − 0.796878i \(-0.706483\pi\)
−0.604141 + 0.796878i \(0.706483\pi\)
\(30\) 0 0
\(31\) 5.98447 1.07484 0.537421 − 0.843314i \(-0.319399\pi\)
0.537421 + 0.843314i \(0.319399\pi\)
\(32\) 5.76897 1.01982
\(33\) 0 0
\(34\) −0.214502 −0.0367867
\(35\) 0 0
\(36\) 0 0
\(37\) 1.00000 0.164399
\(38\) −16.7991 −2.72517
\(39\) 0 0
\(40\) 0 0
\(41\) 0.274494 0.0428688 0.0214344 − 0.999770i \(-0.493177\pi\)
0.0214344 + 0.999770i \(0.493177\pi\)
\(42\) 0 0
\(43\) 7.48354 1.14123 0.570615 − 0.821218i \(-0.306705\pi\)
0.570615 + 0.821218i \(0.306705\pi\)
\(44\) 14.5342 2.19111
\(45\) 0 0
\(46\) 3.23713 0.477289
\(47\) −6.61626 −0.965081 −0.482541 − 0.875874i \(-0.660286\pi\)
−0.482541 + 0.875874i \(0.660286\pi\)
\(48\) 0 0
\(49\) 6.87940 0.982771
\(50\) 0 0
\(51\) 0 0
\(52\) −15.6684 −2.17282
\(53\) −10.4743 −1.43875 −0.719375 − 0.694622i \(-0.755572\pi\)
−0.719375 + 0.694622i \(0.755572\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) −10.3262 −1.37990
\(57\) 0 0
\(58\) 14.8576 1.95089
\(59\) 0.507456 0.0660651 0.0330326 − 0.999454i \(-0.489483\pi\)
0.0330326 + 0.999454i \(0.489483\pi\)
\(60\) 0 0
\(61\) −2.96122 −0.379145 −0.189573 − 0.981867i \(-0.560710\pi\)
−0.189573 + 0.981867i \(0.560710\pi\)
\(62\) −13.6649 −1.73544
\(63\) 0 0
\(64\) −12.9754 −1.62192
\(65\) 0 0
\(66\) 0 0
\(67\) 7.08320 0.865350 0.432675 − 0.901550i \(-0.357570\pi\)
0.432675 + 0.901550i \(0.357570\pi\)
\(68\) 0.301912 0.0366122
\(69\) 0 0
\(70\) 0 0
\(71\) −7.51096 −0.891387 −0.445694 − 0.895186i \(-0.647043\pi\)
−0.445694 + 0.895186i \(0.647043\pi\)
\(72\) 0 0
\(73\) 1.90544 0.223015 0.111507 − 0.993764i \(-0.464432\pi\)
0.111507 + 0.993764i \(0.464432\pi\)
\(74\) −2.28339 −0.265439
\(75\) 0 0
\(76\) 23.6448 2.71224
\(77\) 16.8479 1.92000
\(78\) 0 0
\(79\) 0.537233 0.0604435 0.0302217 − 0.999543i \(-0.490379\pi\)
0.0302217 + 0.999543i \(0.490379\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) −0.626777 −0.0692160
\(83\) 7.52649 0.826140 0.413070 − 0.910699i \(-0.364457\pi\)
0.413070 + 0.910699i \(0.364457\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) −17.0879 −1.84263
\(87\) 0 0
\(88\) −12.5348 −1.33622
\(89\) 0.259584 0.0275158 0.0137579 − 0.999905i \(-0.495621\pi\)
0.0137579 + 0.999905i \(0.495621\pi\)
\(90\) 0 0
\(91\) −18.1627 −1.90397
\(92\) −4.55627 −0.475024
\(93\) 0 0
\(94\) 15.1075 1.55822
\(95\) 0 0
\(96\) 0 0
\(97\) 16.8442 1.71027 0.855133 − 0.518409i \(-0.173475\pi\)
0.855133 + 0.518409i \(0.173475\pi\)
\(98\) −15.7084 −1.58678
\(99\) 0 0
Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 8325.2.a.cj.1.2 6
3.2 odd 2 8325.2.a.ck.1.5 6
5.4 even 2 1665.2.a.u.1.5 yes 6
15.14 odd 2 1665.2.a.t.1.2 ✓ 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1665.2.a.t.1.2 ✓ 6 15.14 odd 2
1665.2.a.u.1.5 yes 6 5.4 even 2
8325.2.a.cj.1.2 6 1.1 even 1 trivial
8325.2.a.ck.1.5 6 3.2 odd 2